When tuning a piano, a musical play a note that should be at 110 Hz while at the same tapping 110 Hz tuning fork and holding it next to the string. He hears beats at 4 Hz.

(A) state and explain what frequencies the piano could be producing
(b) explain how the musician now finishes tuning his piano

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(A) To determine the possible frequencies that the piano could be producing, we can use the concept of beats.

Beats occur when two sound waves with slightly different frequencies interfere with each other. The resulting sound wave has a periodic variation in amplitude known as beats. The rate at which beats occur is equal to the difference between the frequencies of the two sound waves.

In this case, the musician hears beats at 4 Hz, which means there is a difference of 4 Hz between the frequency of the piano note and the tuning fork.

To find the possible frequencies, we can use the formula: Frequency of the piano = Frequency of the tuning fork ± Beats frequency.

1. Piano frequency = Tuning fork frequency + Beats frequency:
- Piano frequency = 110 Hz + 4 Hz = 114 Hz

2. Piano frequency = Tuning fork frequency - Beats frequency:
- Piano frequency = 110 Hz - 4 Hz = 106 Hz

Therefore, the possible frequencies that the piano could be producing are 106 Hz and 114 Hz.

(b) To finish tuning the piano, the musician needs to adjust the tension of the string until the beats disappear. Since the musician hears beats at 4 Hz, the goal is to align the frequency of the piano note with the frequency of the tuning fork.

1. If the piano frequency is 106 Hz, the musician needs to raise the tension by slightly turning the piano tuning pin in the direction that tightens the string. By repeating this process and comparing the sound until the beats disappear, the string will be in tune with the tuning fork.

2. If the piano frequency is 114 Hz, the musician needs to lower the tension by slightly turning the piano tuning pin in the direction that loosens the string. By repeating this process and comparing the sound until the beats disappear, the string will be in tune with the tuning fork.

The musician continues this iterative process, making small adjustments to the string tension until the piano note aligns with the tuning fork frequency and the beats disappear.